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Algebra II Midterm Exam

A free Algebra II lesson from the “Rational Exponents and Radicals” unit, with a worked example and practice problems including step-by-step solutions.

This midterm exam covers the first half of Algebra II: functions (notation, domain/range, inverses, composition, piecewise), polynomial arithmetic, complex numbers, polynomial factorization and division, polynomial graphs, and radicals and rational exponents.

What you'll learn

Why it matters: The midterm is a cumulative checkpoint: it shows whether the function, polynomial, complex-number, and radical tools have become a connected toolkit rather than separate procedures — the transfer the second half of Algebra II and college placement assume.

Worked example

Problem. Solve x^2 - 5x + 6 = 0 by factoring.

  1. Find two numbers that multiply to 6 and add to -5.
  2. -2 and -3 work, so (x - 2)(x - 3) = 0.
  3. Therefore x = 2 or x = 3.

Answer: x = 2 or x = 3

Lesson preview: this page shows 5 practice items. The interactive activity contains 45 practice items and 45 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. Given f(x) = 3x + 2, find f(3).

Show solution
  1. Substitute x = 3.
  2. f(3) = 3(3) + 2 = 9 + 2.
  3. f(3) = 11.

Answer: 11

2. Find the domain of f(x) = 1/(x - 3). Give your answer as the single x-value that must be excluded.

Show solution
  1. The denominator x - 3 cannot equal 0.
  2. Set x - 3 = 0, so x = 3.
  3. Every real number except 3 is allowed, so x = 3 is excluded.

Answer: 3

3. If f(x) = x + 2, find a rule for the inverse f^-1(x).

Show solution
  1. Write y = x + 2, then swap x and y: x = y + 2.
  2. Solve for y: y = x - 2.
  3. So f^-1(x) = x - 2.

Answer: x - 2

4. Let f(x) = 3x + 2 and g(x) = x + 3. Find f(g(3)).

Show solution
  1. First evaluate the inside: g(3) = 3 + 3 = 6.
  2. Now apply f to that result: f(6) = 3(6) + 2 = 18 + 2.
  3. So f(g(3)) = 20.

Answer: 20

5. For f(x) = { 3x + 2, if x < 4; 6x + 8, if x ≥ 4 }, find f(-3).

Show solution
  1. Since -3 < 4, use the rule 3x + 2.
  2. 3(-3) + 2 = -9 + 2 = -7.
  3. So f(-3) = -7.

Answer: -7

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